# Generalized Additive Models (GAM)¶

Generalized Additive Models allow for penalized estimation of smooth terms in generalized linear models.

See Module Reference for commands and arguments.

## Examples¶

The following illustrates a Gaussian and a Poisson regression where categorical variables are treated as linear terms and the effect of two explanatory variables is captured by penalized B-splines. The data is from the automobile dataset https://archive.ics.uci.edu/ml/datasets/automobile We can load a dataframe with selected columns from the unit test module.

In [1]: import statsmodels.api as sm

In [2]: from statsmodels.gam.api import GLMGam, BSplines

# import data
In [3]: from statsmodels.gam.tests.test_penalized import df_autos

# create spline basis for weight and hp
In [4]: x_spline = df_autos[['weight', 'hp']]

In [5]: bs = BSplines(x_spline, df=[12, 10], degree=[3, 3])

# penalization weight
In [6]: alpha = np.array([21833888.8, 6460.38479])

In [7]: gam_bs = GLMGam.from_formula('city_mpg ~ fuel + drive', data=df_autos,
...:                              smoother=bs, alpha=alpha)
...:

In [8]: res_bs = gam_bs.fit()

In [9]: print(res_bs.summary())
Generalized Linear Model Regression Results
==============================================================================
Dep. Variable:               city_mpg   No. Observations:                  203
Model:                         GLMGam   Df Residuals:                   189.13
Model Family:                Gaussian   Df Model:                        12.87
Link Function:               Identity   Scale:                          4.8825
Method:                         PIRLS   Log-Likelihood:                -441.81
Date:                Fri, 14 Jun 2024   Deviance:                       923.45
Time:                        21:41:43   Pearson chi2:                     923.
No. Iterations:                     3   Pseudo R-squ. (CS):             0.9996
Covariance Type:            nonrobust
================================================================================
coef    std err          z      P>|z|      [0.025      0.975]
--------------------------------------------------------------------------------
Intercept       51.9923      1.997     26.034      0.000      48.078      55.906
fuel[T.gas]     -5.8099      0.727     -7.989      0.000      -7.235      -4.385
drive[T.fwd]     1.3910      0.819      1.699      0.089      -0.213       2.995
drive[T.rwd]     1.0638      0.842      1.263      0.207      -0.587       2.715
weight_s0       -3.5556      0.959     -3.707      0.000      -5.436      -1.676
weight_s1       -9.0876      1.750     -5.193      0.000     -12.518      -5.658
weight_s2      -13.0303      1.827     -7.132      0.000     -16.611      -9.450
weight_s3      -14.2641      1.854     -7.695      0.000     -17.897     -10.631
weight_s4      -15.1805      1.892     -8.024      0.000     -18.889     -11.472
weight_s5      -15.9557      1.963     -8.128      0.000     -19.803     -12.108
weight_s6      -16.6297      2.038     -8.161      0.000     -20.624     -12.636
weight_s7      -16.9928      2.045     -8.308      0.000     -21.002     -12.984
weight_s8      -19.3480      2.367     -8.174      0.000     -23.987     -14.709
weight_s9      -20.7978      2.455     -8.472      0.000     -25.609     -15.986
weight_s10     -20.8062      2.443     -8.517      0.000     -25.594     -16.018
hp_s0           -1.4473      0.558     -2.592      0.010      -2.542      -0.353
hp_s1           -3.4228      1.012     -3.381      0.001      -5.407      -1.438
hp_s2           -5.9026      1.251     -4.717      0.000      -8.355      -3.450
hp_s3           -7.2389      1.352     -5.354      0.000      -9.889      -4.589
hp_s4           -9.1052      1.384     -6.581      0.000     -11.817      -6.393
hp_s5           -9.9865      1.525     -6.547      0.000     -12.976      -6.997
hp_s6          -13.3639      2.228     -5.998      0.000     -17.731      -8.997
hp_s7          -13.8902      3.194     -4.349      0.000     -20.150      -7.630
hp_s8          -11.9752      2.556     -4.685      0.000     -16.985      -6.965
================================================================================

# plot smooth components
In [10]: res_bs.plot_partial(0, cpr=True)
Out[10]: <Figure size 640x480 with 1 Axes>

In [11]: res_bs.plot_partial(1, cpr=True)
Out[11]: <Figure size 640x480 with 1 Axes>

In [12]: alpha = np.array([8283989284.5829611, 14628207.58927821])

In [13]: gam_bs = GLMGam.from_formula('city_mpg ~ fuel + drive', data=df_autos,
....:                              smoother=bs, alpha=alpha,
....:                              family=sm.families.Poisson())
....:

In [14]: res_bs = gam_bs.fit()

In [15]: print(res_bs.summary())
Generalized Linear Model Regression Results
==============================================================================
Dep. Variable:               city_mpg   No. Observations:                  203
Model:                         GLMGam   Df Residuals:                   194.75
Model Family:                 Poisson   Df Model:                         7.25
Link Function:                    Log   Scale:                          1.0000
Method:                         PIRLS   Log-Likelihood:                -530.38
Date:                Fri, 14 Jun 2024   Deviance:                       37.569
Time:                        21:41:43   Pearson chi2:                     37.4
No. Iterations:                     6   Pseudo R-squ. (CS):             0.7715
Covariance Type:            nonrobust
================================================================================
coef    std err          z      P>|z|      [0.025      0.975]
--------------------------------------------------------------------------------
Intercept        3.9960      0.130     30.844      0.000       3.742       4.250
fuel[T.gas]     -0.2398      0.057     -4.222      0.000      -0.351      -0.128
drive[T.fwd]     0.0386      0.075      0.513      0.608      -0.109       0.186
drive[T.rwd]     0.0309      0.078      0.395      0.693      -0.122       0.184
weight_s0       -0.0811      0.030     -2.689      0.007      -0.140      -0.022
weight_s1       -0.1938      0.063     -3.067      0.002      -0.318      -0.070
weight_s2       -0.3160      0.082     -3.864      0.000      -0.476      -0.156
weight_s3       -0.3735      0.090     -4.160      0.000      -0.549      -0.198
weight_s4       -0.4187      0.096     -4.360      0.000      -0.607      -0.230
weight_s5       -0.4645      0.103     -4.495      0.000      -0.667      -0.262
weight_s6       -0.5092      0.112     -4.555      0.000      -0.728      -0.290
weight_s7       -0.5469      0.119     -4.598      0.000      -0.780      -0.314
weight_s8       -0.6211      0.137     -4.528      0.000      -0.890      -0.352
weight_s9       -0.6866      0.153     -4.486      0.000      -0.987      -0.387
weight_s10      -0.7370      0.174     -4.228      0.000      -1.079      -0.395
hp_s0           -0.0247      0.010     -2.378      0.017      -0.045      -0.004
hp_s1           -0.0557      0.022     -2.479      0.013      -0.100      -0.012
hp_s2           -0.1046      0.038     -2.719      0.007      -0.180      -0.029
hp_s3           -0.1438      0.050     -2.857      0.004      -0.242      -0.045
hp_s4           -0.1919      0.063     -3.047      0.002      -0.315      -0.068
hp_s5           -0.2567      0.079     -3.231      0.001      -0.412      -0.101
hp_s6           -0.4152      0.120     -3.455      0.001      -0.651      -0.180
hp_s7           -0.4889      0.152     -3.214      0.001      -0.787      -0.191
hp_s8           -0.5470      0.195     -2.810      0.005      -0.928      -0.166
================================================================================

# Optimal penalization weights alpha can be obtained through generalized
# cross-validation or k-fold cross-validation.
# The alpha above are from the unit tests against the R mgcv package.
In [16]: gam_bs.select_penweight()[0]
Out[16]: array([8.2841e+09, 1.4628e+07])

In [17]: gam_bs.select_penweight_kfold()[0]
Out[17]: (398107.1705534969, 79432.82347242805)


### References¶

• Hastie, Trevor, and Robert Tibshirani. 1986. Generalized Additive Models. Statistical Science 1 (3): 297-310.

• Wood, Simon N. 2006. Generalized Additive Models: An Introduction with R. Texts in Statistical Science. Boca Raton, FL: Chapman & Hall/CRC.

• Wood, Simon N. 2017. Generalized Additive Models: An Introduction with R. Second edition. Chapman & Hall/CRC Texts in Statistical Science. Boca Raton: CRC Press/Taylor & Francis Group.

## Module Reference¶

### Model Class¶

 GLMGam(endog[, exog, smoother, alpha, ...]) Generalized Additive Models (GAM) LogitGam(endog, smoother, alpha, *args, **kwargs) Generalized Additive model for discrete Logit

### Results Classes¶

 GLMGamResults(model, params, ...) Results class for generalized additive models, GAM.

### Smooth Basis Functions¶

Currently there is verified support for two spline bases

 BSplines(x, df, degree[, include_intercept, ...]) additive smooth components using B-Splines CyclicCubicSplines(x, df[, constraints, ...]) additive smooth components using cyclic cubic regression splines

statsmodels.gam.smooth_basis includes additional splines and a (global) polynomial smoother basis but those have not been verified yet.

The distribution families in GLMGam are the same as for GLM and so are the corresponding link functions. Current unit tests only cover Gaussian and Poisson, and GLMGam might not work for all options that are available in GLM.

Last update: Jun 14, 2024